Polytopes in arrangements

Boris Aronov, T. K. Dey

    Research output: Contribution to journalArticle

    Abstract

    Consider an arrangement of n hyperplanes in ℝd. Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their maximum combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the past. However, no nontrivial bound was known for the general case where the polytopes may have overlapping interiors, for d > 2. We analyze families of polytopes that do not share vertices. In ℝ3 we show an O (k1/3n2) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in ℝd is Ω (k1/2nd/2) which is a factor of √n away from the best known upper bound in the range nd-2 ≤ k ≤ nd. The case where 1 ≤ k ≤ nd-2 is completely resolved as a known Θ(kn) bound for cells applies here.

    Original languageEnglish (US)
    Pages (from-to)51-63
    Number of pages13
    JournalDiscrete and Computational Geometry
    Volume25
    Issue number1
    StatePublished - 2001

    Fingerprint

    Polytopes
    Arrangement
    Hyperplane
    Pairwise
    Disjoint
    Combinatorial Complexity
    Convex Polytopes
    Cell
    Facet
    Overlapping
    Interior
    Union
    High-dimensional
    Face
    Lower bound
    Upper bound
    Vertex of a graph
    Range of data

    ASJC Scopus subject areas

    • Theoretical Computer Science
    • Computational Theory and Mathematics
    • Discrete Mathematics and Combinatorics
    • Geometry and Topology

    Cite this

    Aronov, B., & Dey, T. K. (2001). Polytopes in arrangements. Discrete and Computational Geometry, 25(1), 51-63.

    Polytopes in arrangements. / Aronov, Boris; Dey, T. K.

    In: Discrete and Computational Geometry, Vol. 25, No. 1, 2001, p. 51-63.

    Research output: Contribution to journalArticle

    Aronov, B & Dey, TK 2001, 'Polytopes in arrangements', Discrete and Computational Geometry, vol. 25, no. 1, pp. 51-63.
    Aronov, Boris ; Dey, T. K. / Polytopes in arrangements. In: Discrete and Computational Geometry. 2001 ; Vol. 25, No. 1. pp. 51-63.
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